Illinois Journal of Mathematics

Naimark’s problem for graph $C^{*}$-algebras

Nishant Suri and Mark Tomforde

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Abstract

Naimark’s problem asks whether a $C^{*}$-algebra that has only one irreducible $*$-representation up to unitary equivalence is isomorphic to the $C^{*}$-algebra of compact operators on some (not necessarily separable) Hilbert space. This problem has been solved in special cases, including separable $C^{*}$-algebras and Type I $C^{*}$-algebras. However, in 2004 Akemann and Weaver used the diamond principle to construct a $C^{*}$-algebra with $\aleph_{1}$ generators that is a counterexample to Naimark’s Problem. More precisely, they showed that the statement “There exists a counterexample to Naimark’s Problem that is generated by $\aleph_{1}$ elements.” is independent of the axioms of ZFC. Whether Naimark’s problem itself is independent of ZFC remains unknown. In this paper, we examine Naimark’s problem in the setting of graph $C^{*}$-algebras, and show that it has an affirmative answer for (not necessarily separable) AF graph $C^{*}$-algebras as well as for $C^{*}$-algebras of graphs in which each vertex emits a countable number of edges.

Article information

Source
Illinois J. Math., Volume 61, Number 3-4 (2017), 479-495.

Dates
Received: 21 September 2017
Revised: 14 May 2018
First available in Project Euclid: 22 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1534924836

Digital Object Identifier
doi:10.1215/ijm/1534924836

Mathematical Reviews number (MathSciNet)
MR3845730

Zentralblatt MATH identifier
06932513

Subjects
Primary: 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]

Citation

Suri, Nishant; Tomforde, Mark. Naimark’s problem for graph $C^{*}$-algebras. Illinois J. Math. 61 (2017), no. 3-4, 479--495. doi:10.1215/ijm/1534924836. https://projecteuclid.org/euclid.ijm/1534924836


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